This paper, I am sorry to say, is deeply sceptical. It is also rather technical. And the scepticalities, I fear, are no easier to renounce than are the techni calities. The paper presents an elementary mathematical result that appears to impinge negatively on any reasonable theory of knowledge with empiricist pretensions. Moreover, it offers nothing positive as a sweetener. This is not because I would not like to endorse something constructive. It is because there is nothing constructive that I would like to endorse. The central problem is this: in what ways and by what standards can we compare two false scientific theories! An eminently plausible sugges tion for quantitative (or numerical) theories will be shown to be useless. This is the suggestion that a quantitative theory B is a better theory than is A at least if it is uniformly more accurate; that is, if the predictions that B makes, given some shared initial conditions, are never farther from the true values (or the observed values) than the predictions that A makes, and sometimes are closer. It will be shown that if an interesting theory B stands in this relation to a theory A then the theory B is true. No false quantitative theory is uniformly more accurate than is any other. I have expressed this crudely, and I have oversimplified enormously. I shall to some extent refine the expression and in some degree under simplify the discussion in the ensuing sections, whose content is here briefly summarized. In Section I, I sketch Popper's theory of verisimilitude, together with recent criticisms of it that are independently due to Pavel Tichy and myself. In Section II, I ask whether Tarski's idea of content can be widened in any useful way, especially in the context of quantitative theories. And in Section III, I wonder whether the accuracy of quantitative theories can be used as an indication of their verisimilitude. Section IV is a short section on terminology and notation. In Section V my main result is proved; Section VI examines essentialism, and the relativization of theories to problems. In Section VII, I look at some of the methodological ramifi